The Liverpool forward opened the scoring midway through the second half with a neat close-range finish at the Borg El Arab Stadium to put them on course for the victory required.
However, Congo levelled with just three minutes of normal time remaining when Arnold Bouka volleyed home.
But there was to be a dramatic twist as Egypt were awarded a spot kick deep into stoppage-time – and Salah kept his nerve to convert and send his country to a World Cup finals for the first time since 1990.